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Finding the set of global minimizers of a piecewise affine function

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Abstract

Coexhausters are families of convex compact sets that allow one to represent the approximation of the increment of a function at a given point in the form of minmax or maxmin of affine functions. We demonstrate that this representation can be used to define a piecewise affine function and therefore coexhausters are a natural technique for studying the problem of finding a global minimum of piecewise affine functions. All the conditions and methods in the current study were obtained by means of coexhausters theory. Firstly, we apply coexhauster based conditions to state and prove necessary and sufficient conditions for a piecewise affine function to be bounded from below. Secondly, we use coexhausters to construct a simple method which allows one to get the minimum value of the studied function and the corresponding set of all its global minimizers. Illustrative numerical examples are provided throughout the paper.

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Acknowledgements

Results in Sect. 4 were obtained in the Institute for Problems in Mechanical Engineering of the Russian Academy of Sciences with the support of Russian Science Foundation (RSF), project No. 20-71-10032. The author sincerely appreciates the anonymous reviewers for the insightful comments and suggestions that helped significantly to improve the quality of the paper.

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Authors and Affiliations

  1. St. Petersburg State University, SPbSU, 7/9 Universitetskaya nab., St. Petersburg, 199034, Russia

    Majid E. Abbasov

  2. Institute for Problems in Mechanical Engineering of the Russian Academy of Sciences, 61, Bolshoj pr. V.O., St. Petersburg, 199178, Russia

    Majid E. Abbasov

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  1. Majid E. Abbasov
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Correspondence to Majid E. Abbasov.

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Abbasov, M.E. Finding the set of global minimizers of a piecewise affine function. J Glob Optim 85, 1–13 (2023). https://doi.org/10.1007/s10898-022-01191-7

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  • DOI: https://doi.org/10.1007/s10898-022-01191-7

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